Generalized hierarchical bases for discontinuous Galerkin discretizations of elliptic problems with highly varying coefficients

In this paper we present a new construction of generalized hierarchical bases (GHB) for symmetric positive definite matrices arising from discontinuous Galerkin discretization of secondorder partial differential equations (PDE) with highly varying coefficients. In the well-established theory of hierarchical basis multilevel methods one basic assumption is that the PDE coefficients are smooth functions on the elements of the coarsest mesh partition. However, as it is shown (for the two-level basis of the scalar elliptic problem), the newly developed GHB yields a robust splitting with respect to jump discontinuities of the PDE coefficient at arbitrary element interfaces on the finest mesh. Though our focus is on a particular family of rotated bilinear finite elements in two space dimensions (2D) here, the proposed rather general approach is neither limited to this particular choice of elements nor to 2D problems. The presented numerical tests are in the spirit of algebraic multilevel iteration (AMLI) methods.